Liquid dispensers, such as liquid soap and sanitizer dispensers, provide a user with a predetermined amount of liquid upon actuation of the dispenser. In addition, it is sometimes desirable to dispense the liquid in the form of foam by, for example, injecting air into the liquid to create a foamy mixture of liquid and air bubbles. Many dispensers are refillable with refill units that comprise a pump (or a pump and an air compressor) and a container. The refills are disposable when the liquid held within the refill unit is emptied. The components of the refill units and pumps are manufactured on one station and then assembled at another station. Accordingly, the more parts utilized in the pump, the more costly the pump is to manufacture.